Charlie's delightful machine
Here is a machine with four coloured lights. Can you develop a strategy to work out the rules controlling each light?
Problem
You may wish to look at Shifting Times Tables before trying this problem.
Charlie's Delightful Machine has four coloured lights. Each light is controlled by a rule.
If you choose a number that satisfies the rule, the light will go on.
Some numbers may turn on more than one light!
Type in some numbers and see which lights you can switch on.
Can you work out how to switch the lights on?
Once you have a strategy, challenge yourself to find some three-digit positive numbers that turn on each light. How about some three-digit negative numbers?
How about a four-digit positive number and a four-digit negative number?
Once you are confident that you can work out the rules for the lights, have a go at A Little Light Thinking, where you can explore how to turn on several lights at once.
Getting Started
You could start by exploring Shifting Times Tables to get a feel for the sequences that turn on the lights.
Try exploring just one light at a time.
Teachers' Resources
Why do this problem?
Many standard questions give exactly the information required to solve them. In this problem, students are encouraged to be curious, to go in search of the information they require, and to work in a systematic way in order to make sense of the results they gather.
The problem could be used to reinforce work on recording and describing linear and quadratic sequences.
Possible approach
This task will require students to have access to computers. If this is not possible, Four Coloured Lights provides students the opportunity to make sense of numerical rules without the need for computers.
- Odd numbers
- Numbers which are 1 more than multiples of 4
- Numbers which are 2 less than multiples of 5
- Numbers which are 3 more than multiples of 7
Level 1 rules are linear sequences of the form $an+b$, with a and b between 2 and 12.
Students could then work in pairs at a computer, trying to light up each of the lights. Challenge them to develop an efficient strategy for working out the rules controlling each light.
While the class is working, note any particularly good ways of recording or working systematically, and highlight them to the rest of the class.
Bring the class together to share insights and conclusions before moving on to A Little Light Thinking, in which students are invited to find sequences that turn several Level 1 lights on simultaneously.
Key questions
Which numbers will you try first?
Can you suggest a number bigger than 1000 that you think will turn on the light?
Possible support
Shifting Times Tables offers students a way of thinking about linear sequences and opportunities to explore how they work.
Students could use a 100 square to record which lights turn on for each number they try.
Possible extension
Level 2 rules are quadratic sequences of the form $an^2+bn+c$ with a=0 or 1
Level 3 rules are quadratic sequences of the form $an^2+bn+c$ with a=0, 0.5, 1, 2 or 3.
Level 3 sequences can be used as a starting point for some detailed exploration into graphical representations of quadratic functions.
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